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Axiom ax-4 1316
Description: Axiom of Specialization. A quantified wff implies the wff without a quantifier (i.e. an instance, or special case, of the generalized wff). In other words if something is true for all x, it is true for any specific x (that would typically occur as a free variable in the wff substituted for φ). (A free variable is one that does not occur in the scope of a quantifier: x and y are both free in x = y, but only x is free in yx = y.) Axiom scheme C5' in [Megill] p. 448 (p. 16 of the preprint). Also appears as Axiom B5 of [Tarski] p. 67 (under his system S2, defined in the last paragraph on p. 77).

Note that the converse of this axiom does not hold in general, but a weaker inference form of the converse holds and is expressed as rule ax-gen 1253. Conditional forms of the converse are given by ax-12 1318, ax-15 1894, ax-16 1553, and ax-17 1331.

Unlike the more general textbook Axiom of Specialization, we cannot choose a variable different from x for the special case. For use, that requires the assistance of equality axioms, and we deal with it later after we introduce the definition of proper substitution - see stdpc4 1520.

The relationship of this axiom to other predicate logic axioms is different than in the classical case. In particular, the current proof of ax4 1892 (which derives ax-4 1316 from certain other axioms) relies on ax-3 716 and so is not valid intuitionistically. (Contributed by NM, 5-Aug-1993.)

Ref Expression
ax-4 (xφφ)

Detailed syntax breakdown of Axiom ax-4
StepHypRef Expression
1 wph . . 3 wff φ
2 vx . . 3 set x
31, 2wal 1250 . 2 wff xφ
43, 1wi 4 1 wff (xφφ)
Colors of variables: wff set class
This axiom is referenced by:  sp  1317  ax-12  1318  hbequid  1322  a4i  1342  a4s  1343  a4sd  1344  hbim  1350  19.3  1358  19.21  1361  19.21bi  1364  hbimd  1376  19.21ht  1381  hbnt  1426  19.12  1437  19.38  1442  ax9o  1456  hbae  1473  equveli  1501  sb2  1512  drex1  1537  ax11b  1565  sbf3t  1585  hbsb4  1586  hbsb4t  1587  a16gb  1605  sb56  1624  sb6  1625  sbalyz  1718  mopick  1799  2eu1  1817  dfsb2  1875  ax5o  1889  ax5  1891  ax11  1895  ax11indalem  1901  ax11inda2ALT  1902
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